Elastic response modeling

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

As of this time, no one has solved the problem of the effect of asperities on a curved surface nor has anyone addressed the issue of crystalline facets. Needless to say, the problem of asperities on an irregular surface has not been addressed. However, Fuller and Tabor [118] have proposed a model that addresses the effects of variations of asperity size on adhesion for the case of planar surfaces. Assuming elastic response to the adhesion-induced stresses, they treated surface roughness as a random series of asperities having a Gaussian height distribution (f> z) and standard deviation o. Accordingly,... 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

The aim for tree breeders and forest managers is to define and grow a plantation which will be elastic in its response to the large stresses induced by high wind speeds. Petty Swain (1985) have established models of the stress-strain responses of forest trees which may be used to define the sizes and morphologies of trees, for a defined range of wind speeds and elastic responses. A typical response of a plantation grown spruce tree to wind speed is shown on Fig. 2. This is a classic stress/strain curve, with an... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

According to the Brice model, for an isovalent series of ions with charge n+ and radius n entering crystal lattice site M, the partition coefficient,, can be described in terms of three parameters (Fig. 3) , the radius of that site the elastic response... 

The experimental determination of RBA, however, is difficult but some attempts have been made and these include direct observation, measurements of electrical conductivity, shrinkage energy, gas adsorption and light scattering. The linear elastic response of paper has been explained in terms of various micromechanical models which take into account both fibre and network properties, including RBA. An example of one which predicts the sheet modulus, Es is given below ... 

Stress-strain relationships for soil are difficult to model due to their complexity. In normal practice, response of soil consists of analyzing compression and shear stresses produced by the structure, applied as static loads. Change in soil strength with deformation is usually disregarded. Clay soils will exhibit some elastic response and are capable of absorbing blast-energy however, there may be insufficient test data to define this response quantitatively. Soil has a very low tensile capacity thus the stress-strain relationship is radically different in the tension region than in compression. 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

Voigt-Kelvin model or element Model consisting of an ideal spring and dashpot in parallel in which the elastic response is retarded by viscous resistance of the fluid in the dashpot. 

In which element or model for a viscoelastic body will the elastic response be retarded by viscous resistance (a) Maxwell or (b) Voigt-Kelvin ... 

As the term implies, viscoelasticity is the response of a material to an applied stress that has both a viscous and an elastic component. In addition to a recoverable elastic response to an applied force, polymers can undergo permanent deformation at high strains, just as was the case for metals and some glasses, as described previously. The mechanism of permanent deformation is different in polymers, however, and can resemble liquid-like, or viscous flow, just like we described in Chapter 4. Let us first develop two important theoretical models to describe viscoelasticity, then describe how certain polymers exhibit this important property. 

The creep parameters J and y are obtained through transverse creep experiments. The initial compliance is the elastic response of the material (Equation 8.41). In general, the creep parameters J and y, and the shift factor aT may all be dependent on the cure state of the material. For the current process model the shift factor is assumed to be separable and, as such, is only temperature dependent. As a first approximation the creep parameters are represented as linear functions of the degree of cure. ... 

Using planar tetrakaidecahedra as the model, on the other hand, causes the square faces to shrink to zero area at the yield point. The unit cell therefore resembles a true rhombic dodecahedron. The elastic response was found to be anisotropic (i.e. dependent on initial cell orientation) for the planar model, up to the elastic limit. This is in contrast to the monodisperse 2D case, which is... 

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

The parameters y and e relate to the model of nephron-nephron interaction to be discussed in Section 12.5. In order to represent the hemodynamic coupling it is necessary to also introduce a parameter Cgj0 that describes the elastic response of... 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

The Voigt-Kelvin element (retarded elastic response), represented by a spring and a dashpot in parallel. The elastic response is not instantaneous but retarded by a viscous resistance. The two contributions to the stress are additive in this model whereas the strains are equal ... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

The first thing we want to model is a simple linear elastic response, as described by Hooke s law. The equation for simple uniaxial extension is shown in Figure 13-87,... 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

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